Bannister, Joshua;
(2024)
Acoustic scattering by fractal inhomogeneities.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
In this thesis, we present the first rigorous numerical analysis of acoustic scattering by fractal inhomogeneities. Our methodology is to reformulate the heterogeneous Helmholtz equation into the Lippmann-Schwinger equation (LSE) on the inhomogeneity and to solve the LSE numerically. This reformulation is equivalent provided the boundary has zero two-dimensional Lebesgue measure, such as for the Koch snowflake, which we study extensively. Indeed, for the snowflake, we analyse two distinct numerical methods. The first, known as the “pre-fractal" method, replaces the snowflake with a sequence of smoother pre-fractal sets, on which conventional numerical methods can be applied, whilst the second, known as the “IFS method", utilises the self-similarity of the snowflake to produce a quasi-uniform mesh composed of rotated and scaled copies of the snowflake, referred to as an Lh-mesh, on which a piecewise constant Galerkin FEM is applied. We show that both methods converge, are quasi-optimal, and are stable, but the pre-fractal method converges sub-optimally while the IFS method converges optimally. We then conduct a fully discrete error analysis of the IFS method for a particular scattering problem applied to the snowflake. We then show that each Lh-mesh of the snowflake admits lattice structures, which, in conjunction with the structure of the LSE, allow us to apply a modification of the classical circulant embedding method to compress the resulting Galerkin matrix and accelerate iterative solutions of the Galerkin equations via the fast Fourier transform. We then use this technique to implement the IFS method, validating our fully discrete error analysis. Finally, we apply the methods we developed for the Koch snowflake to further examples, namely the Gosper Island and the Fudgeflake, and present numerical results suggesting much of our analysis holds beyond the Koch snowflake.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Acoustic scattering by fractal inhomogeneities |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2024. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/).Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| Keywords: | Fractal, Numerical analysis, integral equations, applied analysis, Numerical linear algebra |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10198108 |
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